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  • j. differential geometry

    56 (2000) 167-188



    AbstractA higher dimensional analogue of Kodairas canonical bundle formula isobtained. As applications, we prove that the log-canonical ring of a kltpair with 3 is nitely generated, and that there exists an eectivelycomputable natural number M such that |MKX | induces the Iitaka beringfor every algebraic threefold X with Kodaira dimension = 1.

    1. Introduction

    If f : X C is a minimal elliptic surface over C, then the relativecanonical divisor KX/C is expressed as

    KX/C = fL+


    mP 1mP

    f(P ),(1)

    where L is a nef divisor on C and P runs over the set of points such thatf(P ) is a multiple ber with multiplicity mP > 1. It is the key in theestimates on the plurigenera Pn(X) that the coecients (mP 1)/mPare close to 1 [12]. Furthermore 12L is expressed as

    12KX/C = fjOP(1) + 12


    mP 1mP

    f(P ) +


    where Q is an integer [0, 12) and j : C P1 is the j-function [5,(2.9)]. The computation of these coecients is based on the explicitclassication of the singular bers of f , which made the generalizationdicult.

    1991 Mathematics Subject Classication. 14N30; 14E30, 14J40.Received October 10, 2000.


  • 168 osamu fujino & shigefumi mori

    We note that L in the exact analogue of the formula (1) for the casedimX/C = 2 need not be a divisor (Example 2.7) and that the formula(2) is more natural to look at if L is allowed to be a Q-divisor.

    The higher dimensional analogue of the formula (2) is treated inSection 2 as a renement of [15, 5 Part II] and the log version inSection 4. We give the full formula only in 4.5 to avoid repetition. Theestimates of the coecients are treated in 2.8, 3.1 and 4.5. (See 3.9 onthe comparison of the formula (2) and our estimates.) We note that thecoecients in the formula (2) are of the form 1 1/m except for thenite number of exceptions 1/12, , 11/12. In the generalized formula,the coecients are in a more general form (cf. 4.5.(v)), which still enjoysthe DCC (Descending Chain Condition) property of Shokurov.

    The following are some of the applications.

    1. (Corollary 5.3) If (X,) is a klt pair with (X,KX+) 3, thenits log-canonical ring is nitely generated.

    2. (Corollary 6.2) There exists an eectively computable naturalnumber M such that |MKX | induces the Iitaka bering for ev-ery algebraic threefold X with Kodaira dimension (X) = 1.

    To get the analogue for an (m + 1)-dimensional X (m 3) with(X) = 1, it remains to show that an arbitrary m-fold F with (F ) = 0and pg(F ) = 1 is birational to a smooth projective model with eectivelybounded m-th Betti number.

    Notation. Let Z>0 (resp. Z0) be the set of positive (resp. non-negative) integers. We work over C in this note. Let X be a normalvariety and B,B Q-divisors on X.

    If B B is eective, we write B B or B B.We write B B if B B is a principal divisor on X (linear

    equivalence of Q-divisors).Let B+, B be the eective Q-divisors on X without common ir-

    reducible components such that B+ B = B. They are called thepositive and the negative parts of B.

    Let f : X C be a surjective morphism. Let Bh, Bv be the Q-divisors on X with Bh + Bv = B such that an irreducible componentof Supp B is contained in Supp Bh i it is mapped onto C. They arecalled the horizontal and the vertical parts of B over C. B is said tobe horizontal (resp. vertical ) over C if B = Bh (resp. B = Bv). Thephrase over C might be suppressed if there is no danger of confusion.

  • a canonical bundle formula 169

    As for other notions, we mostly follow [14]. However we introduce aslightly dierent terminology to distinguish the pairs with non-eectiveboundaries (cf. [10]).

    A pair (X,D) consists of a normal variety X and a Q-divisor D. IfKX + D is Q-Cartier, we can pull it back by an arbitrary resolutionf : Y X and obtain the formula

    KY = f(KX +D) +i


    where Ei are prime divisors and ai Q. The pair (X,D) is said to besub klt (resp. sub lc) if ai > 1 (resp. 1) for every resolution f andevery i. Furthermore, (X,D) is said to be klt (resp. lc) if D is eective.

    Acknowledgements. This note is an expanded version of the sec-ond authors lecture On a canonical bundle formula at Algebraic Ge-ometry Workshop in Hokkaido University June 1994.

    The authors were partially supported by the Grant-in-Aid for Scien-tic Research, the Ministry of Education, Science, Sports and Culture ofJapan. The second author was also partially supported by the InamoriFoundation.

    We would like to thank Professors H. Clemens and J. Kollar forhelpful conversations and Professor K. Ohno for pointing out mistakesin an earlier version.

    2. Semistable part of KX/C

    In this section, we rene the results of [15, 5, Part II] after puttingthe basic results together.

    2.1. Let f : X C be a surjective morphism of a normalprojective variety X of dimension n = m+ l to a nonsingular projectivel-fold C such that

    (i) X has only canonical singularities, and(ii) the generic ber F of f is a geometrically irreducible variety

    with Kodaira dimension (F ) = 0. We x the smallest b Z>0 suchthat the b-th plurigenus Pb(F ) is non-zero.

    Proposition 2.2. There exists one and only one Q-divisor Dmodulo linear equivalence on C with a graded OC-algebra isomorphism

    i0O(iD) = i0(fO(ibKX/C)),

  • 170 osamu fujino & shigefumi mori

    where M denotes the double dual of M .Furthermore, the above isomorphism induces the equality

    bKX = f(bKC +D) +B,

    where B is a Q-divisor on X such that fOX(iB+) = OC (i > 0)and codim(f(Supp B) C) 2. We note that for an arbitrary openset U of C, D|U and B|f1(U) depend only on f |f1(U).

    Proof. By [15, (2.6.i)], there exists c > 0 such that

    (fO(ibcKX/C)) = {(fO(bcKX/C))}i (i > 0).

    Choose an embedding : (fO(bKX/C)) Q(C) into the functioneld of C, and we can dene a Weil divisor cD by

    c : (fO(bcKX/C)) = O(cD) Q(C).

    D modulo linear equivalence does not depend on the choice of .Since taking the double dual has no eect on codimension 1 points,

    there is a natural inclusion

    fOC(cD) OX(bcKX/C) on X \ f1(some codim 2 subset of C).

    Extending it toX, we obtain a Q-divisor B such that B = bKX/CfD.It is easy to see that B satises the required conditions. q.e.d.

    Denition 2.3. Under the notation of 2.2, we denote D by LX/C .It is obvious that LX/C depends only on the birational equivalence classof X over C.

    If X has bad singularities, then we take a nonsingular model X ofX and use LX/C as our denition of LX/C .

    Proposition 2.4 (Viehweg). Let : C C be a nite surjec-tive morphism from a nonsingular l-fold C and let f : X C be anonsingular model of X C C C . Then there is a natural relation

    LX/C LX/C .

    Furthermore if X C C has a semistable resolution over a neighbor-hood of a codimension 1 point P of C , or if X(P ) has only canonicalsingularities, then P Supp(LX/C LX/C).

  • a canonical bundle formula 171

    Proof. Except for the last assertion, this is due to [22, 3] (cf. [15,(4.10)]). If X(P ) has only canonical singularities, then X C C hasonly canonical singularities in a neighborhood of f 1(P ) by [20, Propo-sition 7] or [9] because so does the generic ber F of f . Thus 2.4 follows.


    Corollary 2.5. There exists one and only one Q-divisorLssX/C ( LX/C) such that

    (i) LssX/C LX/C for arbitrary : C C as in 2.4, and

    (ii) LssX/C = LX/C at P if in 2.4 is such that X C C C

    has a semistable resolution X C over a neighborhood of P orX(P ) has only canonical singularities.

    There exists an eective divisor C such that every birational mor-phism : C C from a nonsingular projective l-fold with () ansnc divisor has the following property: Let X be a projective resolutionof X C C and f : X C the induced morphism. Then LssX/C isnef.

    Proof. When C /C is Galois with group G, LX/C is G-invariant and

    therefore descends to a Q-subdivisor of LX/C . The minimum LssX/C (LX/C) of all the descents exists by 2.4, whence the uniqueness follows.

    The last assertion is proved in [15, 5, part II] though it is not explic-itly stated there. In [15, (5.13)], First our is a divisor containing thediscriminant locus of f and h constructed in the proof of [15, (5.15.2)](which are f and h in our 2.6). Then [15, (5.14.1)] shows that ourLssX/C/b is equal to Pf ,bc for suciently divisible c Z>0. Finally [15,(5.15.3)] shows that Pf ,bc is nef. q.e.d.

    Remark 2.6. Under the notation of 2.1, consider the followingconstruction. Since dim |bKF | = 0, there exists a Weil divisor W on Xsuch that

    (i) W h is eective and fO(iW h) = OC for all i > 0, and(ii) bKX W is a principal divisor () for some non-zero rational

    function on X.

    Let s : Z X be the normalization of X in Q(X)(1/b). Then theabove proof actually shows the following:

  • 172 osamu fujino & shigefumi mori

    Fix resolutions X and Z of X and Z. We write f : X C and h : Z C. Then as the divisor C in 2.5,we can take an arbitrary eective divisor C such thatf : X C and h : Z C are smooth over C \ .

    Example 2.7. Let F be a K3 surface with a free involution :F F so that E = F/{1, } is an Enriques surface. Let j : P1 P1 bethe involution x x, so that 0 and are the only xed points. Letf : X = P1 F/{1, j } C = P1/{1, j} be the map induced by therst projection. Then f is smooth over C \ {0,}, and f(0) = 2E0and f() = 2E, where E0 E E. Using KEt KX +Et|Et 0for t = 0 and , one easily sees that KX/C fO(1) and that LssX/C =12(0) 12(). Thus LssX/C is only a Q-divisor tting in the analogue ofthe formula (1):